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Theorem dftpos2 8178
Description: Alternate definition of tpos when 𝐹 has relational domain. (Contributed by Mario Carneiro, 10-Sep-2015.)
Assertion
Ref Expression
dftpos2 (Rel dom 𝐹 → tpos 𝐹 = (𝐹 ∘ (𝑥dom 𝐹 {𝑥})))
Distinct variable group:   𝑥,𝐹

Proof of Theorem dftpos2
StepHypRef Expression
1 dmtpos 8173 . . 3 (Rel dom 𝐹 → dom tpos 𝐹 = dom 𝐹)
21reseq2d 5941 . 2 (Rel dom 𝐹 → (tpos 𝐹 ↾ dom tpos 𝐹) = (tpos 𝐹dom 𝐹))
3 reltpos 8166 . . 3 Rel tpos 𝐹
4 resdm 5986 . . 3 (Rel tpos 𝐹 → (tpos 𝐹 ↾ dom tpos 𝐹) = tpos 𝐹)
53, 4ax-mp 5 . 2 (tpos 𝐹 ↾ dom tpos 𝐹) = tpos 𝐹
6 df-tpos 8161 . . . 4 tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}))
76reseq1i 5937 . . 3 (tpos 𝐹dom 𝐹) = ((𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})) ↾ dom 𝐹)
8 resco 6206 . . 3 ((𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})) ↾ dom 𝐹) = (𝐹 ∘ ((𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) ↾ dom 𝐹))
9 ssun1 4136 . . . . 5 dom 𝐹 ⊆ (dom 𝐹 ∪ {∅})
10 resmpt 5995 . . . . 5 (dom 𝐹 ⊆ (dom 𝐹 ∪ {∅}) → ((𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) ↾ dom 𝐹) = (𝑥dom 𝐹 {𝑥}))
119, 10ax-mp 5 . . . 4 ((𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) ↾ dom 𝐹) = (𝑥dom 𝐹 {𝑥})
1211coeq2i 5820 . . 3 (𝐹 ∘ ((𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) ↾ dom 𝐹)) = (𝐹 ∘ (𝑥dom 𝐹 {𝑥}))
137, 8, 123eqtri 2765 . 2 (tpos 𝐹dom 𝐹) = (𝐹 ∘ (𝑥dom 𝐹 {𝑥}))
142, 5, 133eqtr3g 2796 1 (Rel dom 𝐹 → tpos 𝐹 = (𝐹 ∘ (𝑥dom 𝐹 {𝑥})))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  cun 3912  wss 3914  c0 4286  {csn 4590   cuni 4869  cmpt 5192  ccnv 5636  dom cdm 5637  cres 5639  ccom 5641  Rel wrel 5642  tpos ctpos 8160
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-fv 6508  df-tpos 8161
This theorem is referenced by:  tposf12  8186
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